New almost sure convergence results are developed for a special form of the multidimensional Robbins-Monro (RM) stochastic approximation procedure. The special form treated can be viewed as a stochastic approximation to the solution w = w sub o epsilon Rp of the linear equations Rw = P, where R is a pxp positive definite symmetric matrix. This special form commonly arises in adaptive signal processing applications. Essentially, previous convergence results for the RM procedure contain a common 'conditional expectation condition' which is extremely difficult (if not impossible) to satisfy when the 'training data' is a correlated sequence. In contrast, the new convergence results incorporate moment conditions and covariance function decay rate conditions. The ease with which these results can be applied in many cases is illustrated.